Theorem txindis 23770

Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txindis	⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Proof of Theorem txindis

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 4305	. . . . . . 7 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
2		indistop 23138	. . . . . . . . . . 11 ⊢ {∅, 𝐴} ∈ Top
3		indistop 23138	. . . . . . . . . . 11 ⊢ {∅, 𝐵} ∈ Top
4		eltx 23704	. . . . . . . . . . 11 ⊢ (({∅, 𝐴} ∈ Top ∧ {∅, 𝐵} ∈ Top) → (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
5	2, 3, 4	mp2an 704	. . . . . . . . . 10 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
6		rsp 3251	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
7	5, 6	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
8		elssuni 4903	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ ∪ ({∅, 𝐴} ×t {∅, 𝐵}))
9		indisuni 23139	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐴) = ∪ {∅, 𝐴}
10		indisuni 23139	. . . . . . . . . . . . . . 15 ⊢ ( I ‘𝐵) = ∪ {∅, 𝐵}
11	2, 3, 9, 10	txunii 23729	. . . . . . . . . . . . . 14 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ∪ ({∅, 𝐴} ×t {∅, 𝐵})
12	8, 11	sseqtrrdi 3977	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
13	12	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
14		ne0i 4293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝑧 × 𝑤) → (𝑧 × 𝑤) ≠ ∅)
15	14	ad2antrl 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ≠ ∅)
16		xpnz 6156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅) ↔ (𝑧 × 𝑤) ≠ ∅)
17	15, 16	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅))
18	17	simpld 499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ≠ ∅)
19	18	neneqd 2961	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑧 = ∅)
20		simpll 778	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, 𝐴})
21		indislem 23136	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐴)} = {∅, 𝐴}
22	20, 21	eleqtrrdi 2872	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, ( I ‘𝐴)})
23		elpri 4612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {∅, ( I ‘𝐴)} → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
24	22, 23	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
25	24	ord 877	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑧 = ∅ → 𝑧 = ( I ‘𝐴)))
26	19, 25	mpd 16	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 = ( I ‘𝐴))
27	17	simprd 500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ≠ ∅)
28	27	neneqd 2961	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑤 = ∅)
29		simplr 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, 𝐵})
30		indislem 23136	. . . . . . . . . . . . . . . . . . 19 ⊢ {∅, ( I ‘𝐵)} = {∅, 𝐵}
31	29, 30	eleqtrrdi 2872	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, ( I ‘𝐵)})
32		elpri 4612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ {∅, ( I ‘𝐵)} → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
33	31, 32	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
34	33	ord 877	. . . . . . . . . . . . . . . 16 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑤 = ∅ → 𝑤 = ( I ‘𝐵)))
35	28, 34	mpd 16	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 = ( I ‘𝐵))
36	26, 35	xpeq12d 5692	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) = (( I ‘𝐴) × ( I ‘𝐵)))
37		simprr 784	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ⊆ 𝑥)
38	36, 37	eqsstrrd 3971	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
39	38	adantll 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
40	13, 39	eqssd 3953	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵)))
41	40	ex 417	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
42	41	rexlimdvva 3220	. . . . . . . . 9 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
43	7, 42	syld 48	. . . . . . . 8 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
44	43	exlimdv 1961	. . . . . . 7 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑦 𝑦 ∈ 𝑥 → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
45	1, 44	biimtrid 245	. . . . . 6 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (¬ 𝑥 = ∅ → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
46	45	orrd 876	. . . . 5 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
47		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
48	47	elpr 4613	. . . . 5 ⊢ (𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ↔ (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
49	46, 48	sylibr 237	. . . 4 ⊢ (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))})
50	49	ssriv 3940	. . 3 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ {∅, (( I ‘𝐴) × ( I ‘𝐵))}
51	9	toptopon 23053	. . . . . . 7 ⊢ ({∅, 𝐴} ∈ Top ↔ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)))
52	2, 51	mpbi 233	. . . . . 6 ⊢ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
53	10	toptopon 23053	. . . . . . 7 ⊢ ({∅, 𝐵} ∈ Top ↔ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵)))
54	3, 53	mpbi 233	. . . . . 6 ⊢ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))
55		txtopon 23727	. . . . . 6 ⊢ (({∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) ∧ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))) → ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))))
56	52, 54, 55	mp2an 704	. . . . 5 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵)))
57		topgele 23066	. . . . 5 ⊢ (({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))) → ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵))))
58	56, 57	ax-mp 5	. . . 4 ⊢ ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵)))
59	58	simpli 488	. . 3 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵})
60	50, 59	eqssi 3952	. 2 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (( I ‘𝐴) × ( I ‘𝐵))}
61		txindislem 23769	. . 3 ⊢ (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
62	61	preq2i 4702	. 2 ⊢ {∅, (( I ‘𝐴) × ( I ‘𝐵))} = {∅, ( I ‘(𝐴 × 𝐵))}
63		indislem 23136	. 2 ⊢ {∅, ( I ‘(𝐴 × 𝐵))} = {∅, (𝐴 × 𝐵)}
64	60, 62, 63	3eqtri 2788	1 ⊢ ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4561  {cpr 4590  ∪ cuni 4871   I cid 5555   × cxp 5659  ‘cfv 6536  (class class class)co 7410  Topctop 23029  TopOnctopon 23046   ×_t ctx 23696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-topgen 17495  df-top 23030  df-topon 23047  df-bases 23082  df-tx 23698

This theorem is referenced by: (None)